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Topic: Another Sum-Product Variant (Read 660 times) |
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ThudnBlunder
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Another Sum-Product Variant
« on: Jun 24th, 2008, 3:38pm » |
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Three people V, C, and X are joined by M who holds sixteen playing cards which they can see:
SPADES: J, 8, 7, 4, 3, 2
DIAMONDS: A, 5
CLUBS: K, Q, 6 ,5, 4
HEARTS: A, Q, 4
M secretly selects a card at random and tells V its value and C its colour.
In X's hearing, he then asks them if they know which card he chose.
The following conversation ensues:
V: I don't know what the card is.
C: I knew that you didn't know.
V: I know the card now.
C: I know it, too.
X then thinks for a moment and correctly deduces what M's card is.
How is this possible?
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| « Last Edit: Jun 24th, 2008, 7:03pm by ThudnBlunder » |
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SMQ
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Re: Another Sum-Product Variant
« Reply #1 on: Jun 24th, 2008, 6:51pm » |
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From each statement X (and we) can deduce:
V: The card is a 4, 5, Q, or A (because the others only appear in a single suit)
C: The card is a Diamond or Heart (because the other suits have unique cards)
V: The card is not an Ace (because an Ace appears in both Diamonds and Hearts)
C: The card is the Five of Diamonds (because there are still two possible Hearts)
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Eigenray
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Re: Another Sum-Product Variant
« Reply #2 on: Jun 25th, 2008, 1:45am » |
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Colour == suit?
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SMQ
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Re: Another Sum-Product Variant
« Reply #3 on: Jun 25th, 2008, 5:33am » |
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on Jun 25th, 2008, 1:45am, Eigenray wrote:
No, just color. Does that pose a problem for my above reasoning?
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Eigenray
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Re: Another Sum-Product Variant
« Reply #4 on: Jun 25th, 2008, 5:46am » |
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Then I don't see how C is able to determine the card.
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| « Last Edit: Jun 25th, 2008, 5:47am by Eigenray » |
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rmsgrey
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Re: Another Sum-Product Variant
« Reply #5 on: Jun 25th, 2008, 8:11am » |
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on Jun 25th, 2008, 5:46am, Eigenray wrote:| Then I don't see how C is able to determine the card. |
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Me neither:
4H, 5D and QH all look to be possible given the information available - we know, and C knows, that the card is red (were it black, C couldn't have known that V didn't know its value) and, from V being able to identify it from its colour and value, we know it's not an Ace, but I don't see any way to narrow down V's knowledge further than 4,5,Q
Of course, if C knows the suit, rather than just the colour, we can get 5D out of it all.
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ThudnBlunder
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Re: Another Sum-Product Variant
« Reply #6 on: Jun 25th, 2008, 6:00pm » |
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on Jun 25th, 2008, 5:46am, Eigenray wrote:| Then I don't see how C is able to determine the card. |
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I agree. But the book does say 'colour', not 'suit'. As 'suit' allows X to deduce 5D, I guess the book must be wrong. (No answer is given.)
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| « Last Edit: Jun 25th, 2008, 9:34pm by ThudnBlunder » |
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